Reference request for characteristic subgroups of free abelian groups

Question

The characteristic subgroups of a free abelian group $F$ are the $k F$ with $k \in \mathbb{N}$. (In particular, they are verbal and hence fully invariant.) I know a proof when $F$ has finite rank, but I would like to know a reference of this fact in the literature, which has the proof. (In a paper I am working on I prefer to not write down the proof.)

Sketch of the proof when $F$ has finite rank, say $F=\mathbb{Z}^n$: one knows that $u,v \in \mathbb{Z}^n$ are conjugated under $\mathrm{Aut}(\mathbb{Z}^n)$ iff $\mathrm{gcd}(u_1,\dotsc,u_n) = \mathrm{gcd}(v_1,\dotsc,v_n)$. If $U \subseteq \mathbb{Z}^n$ is characteristic and $u \in U$ has $\mathrm{gcd}(u_1,\dotsc,u_n) = k$, it follows $k e_1,\dotsc,k e_n \subseteq U$ and hence $k \mathbb{Z}^n \subseteq U$. Now $\{k \in \mathbb{Z}: k \mathbb{Z}^n \subseteq U\}$ is a subgroup, and a generator of this subgroup does the job.

Answer 1

EDIT: The results in this answer appear to be insufficient to qualify as a proper answer. I'm leaving this answer up as it is too long for a comment and possibly of interest to anyone interested in this question.

This result is mentioned as "well known" (without giving a reference, sadly), and used to prove a stronger result in the following paper. A free, legal scan is available here.

Göbel, Rüdiger, The characteristic subgroups of the Baer-Specker group, Math. Z. 140, 289-292 (1974). ZBL0281.20026.

The introduction is pretty self-explanatory:

It is well known, that the only characteristic subgroups of a free abelian group $A$ are the subgroups $n \cdot A$ for all integers $n \geqq 0$. We will show that this is true for a larger class of groups, namely for all $\mathbf{K}$-direct sums $\bigoplus_{\mathbf{K}} \mathbb{Z}$ of infinite cyclic groups $\mathbb{Z}$, where $\mathbf{K}$ is an ideal of an arbitrary power set. In particular it is true for arbitrary cartesian powers $\mathbb{Z}^{\mathbf{I}}$.

Alternatively, GroupProps has the following two articles:

• Verbal subgroup equals power subgroup in abelian group,
• Characteristic equals verbal in free abelian group.

Combining these two results gives you what you want, though unfortunately the latter article doesn't include a proof. Moreover, no references are given in these articles, nor was I able to find these results anywhere else.

Answer 2

Your proof for finite rank free abelian groups works almost verbatim for general free abelian groups.

Proof of statement for general free abelian groups:

Let $U\subseteq \oplus_{i\in I} \mathbb{Z}$ be a characteristic subgroup, where $I$ is some (possibly infinite) set. Let $e_i$, $i\in I$ generate the summands of $\oplus_{i\in I} \mathbb{Z}$. For $u \in U$ let $\gcd(u)$ denote the greatest common divisor of the coefficients on the the $e_i$ for $u$.

Any $u\in U$ satisfies $$u\in \oplus_{i\in A} \mathbb{Z}\subseteq \oplus_{i\in I} \mathbb{Z},$$ where $A$ is a finite subset of $I$, and the inclusion of groups is induced by the inclusion of sets $A\subseteq I$.

Then by encoding the steps in Euclid's algorithm applied to $u$ as matrices, we obtain an automorphism of $\oplus_{i\in A} \mathbb{Z}$ which maps $$u\mapsto gcd(u)e_j,$$ for some $j\in A$. This automorphism extends to an automorphism of $\oplus_{i\in I} \mathbb{Z}$, using $I=A\sqcup (I\backslash A)$. Then for any $k\in I$, some permutation of $I$ will map $j\mapsto k$, inducing an automorphism of $\oplus_{i\in I} \mathbb{Z}$, mapping $e_j\mapsto e_k$. Composing these automorphisms, we map $$u\mapsto \gcd(u)e_k.$$

Thus for all $u\in U$, we have $\gcd(u)\left(\oplus_{i\in I} \mathbb{Z}\right)\subseteq U$. Thus $$\gcd\{\gcd(u)|u\in U\}\left(\oplus_{i\in I} \mathbb{Z}\right)\subseteq U.$$

Conversely, given $u\in U$, we have $$\gcd\{\gcd(u)|u\in U\}\,\,|\,\, \gcd(u),$$ so $$U\subseteq \gcd\{\gcd(u)|u\in U\}\left(\oplus_{i\in I} \mathbb{Z}\right).$$